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A087981
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E.g.f.: exp(-2*x) / (1-x)^2.
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18
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1, 0, 2, 4, 24, 128, 880, 6816, 60032, 589312, 6384384, 75630080, 972387328, 13483769856, 200571078656, 3185540657152, 53800242216960, 962741176500224, 18195808235880448, 362183230599856128, 7572922094360723456, 165945771111208714240, 3802923921298533384192, 90965940197460917878784, 2267151124921333646884864
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OFFSET
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0,3
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COMMENTS
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Permanent of an (n+1) X (n+1) (+1, -1)-matrix with exactly n -1's on the diagonal and 1's everywhere else.
It is conjectured by Kräuter and Seifter that for n >= 5 a(n-1) is the maximal possible value for the permanent of a nonsingular n X n (+1, -1)-matrix. I do not know for which values of n this has been confirmed - compare A087982. - N. J. A. Sloane
The Kräuter conjecture on permanents is true (see Budrevich and Guterman). - Sergei Shteiner, Jan 17 2020
The maximal possible value for the permanent of a singular n X n (+1, -1)-matrix is obviously n!.
Degree of the "hyperdeterminant" of a multilinear polynomial on (\P^1(\C))^n, or equivalently of an element of (\C^2)^{⊗ n}: see Gelfand, Kapranov and Zelevinsky. - Eric Rains, Mar 15 2004
a(n) is the number of n X n (-1, 0, 1)-matrices containing in every row and every column exactly one -1 and one 1 such that the main diagonal does not contain 0's. - Vladimir Shevelev, Apr 01 2010
a(n) is the number of colored permutations with no fixed points of n elements where each cycle is one of two colors. - Michael Somos, Jan 19 2011
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REFERENCES
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I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Birkhauser, 1994; see Corollary 2.10 in Chapter 14 (p. 457).
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LINKS
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I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky, Hyperdeterminants, Advances in Mathematics 96(2) (1992), 226-263; see Corollary 3.10 (p. 246).
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FORMULA
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Krauter and Seifter prove that the permanent of an n X n {-1, 1} matrix is divisible by 2^{n - [log_2(n)] - 1}.
Let c(n) be the permanent of the {-1, +1}-matrix of order n X n with n diagonal -1's only. Let a(n) be the permanent of the {-1, +1}-matrix of order (n+1) X (n+1) with n diagonal -1's only. Then by expanding along the first row (like determinant, but with no sign) we get c(n+1) = -c(n) + n a(n-1), a(n) = c(n) + n a(n-1), with c(2) = 2, a(2) = 2. {c(n)} has e.g.f. exp(-2x)/(1-x), see A000023. Also a(n) = c(n+1) + 2*c(n).
The following 4 formulas hold: a(n) = Sum_{k = 0..n} C(n, k)*D_k*D_{n-k}, where D_n = A000166(n); a(n) = n!*Sum_{j = 0..n} (n+1-j)*(-2)^j/j!; a(0) = 1, a(1) = 0 and, for n > 0, a(n+1) = n*(a(n) + 2*a(n-1)); a(0) = 1 and, for n > 0, a(n) = (n*(n+3)/(n+2))*a(n-1) - (-2)^(n+1)/(n+2). - Vladimir Shevelev, Apr 01 2010 [edited by Michael Somos, Jan 19 2011]
G.f.: 1/(1-2x^2/(1-2x-6x^2/(1-4x-12x^2/(1-6x-20x^2/(1-.../(1-2n*x-(n+1)(n+2)x^2/(1-... (continued fraction). - Paul Barry, Apr 11 2011
E.g.f.: 1/U(0) where U(k)= 1 - 2*x/( 1 + x/(2 - x - 4/( 2 + x*(k+1)/U(k+1)))) ; (continued fraction, 3rd kind, 4-step). - Sergei N. Gladkovskii, Oct 28 2012
G.f.: 1/Q(0), where Q(k) = 1 + 2*x - x*(k+2)/(1 - x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 22 2013
G.f.: 1/Q(0) where Q(k) = 1 - 2*k*x - x^2*(k + 1)*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 10 2013
G.f.: S(x)/x - 1/x = G(0)/x - 1/x, where S(x) = sum(k >= 0, k!*(x/(1+2*x))^k ), G(k) = 1 + (2*k + 1)*x/( 1+2*x - 2*x*(1+2*x)*(k+1)/(2*x*(k+1) + (1+2*x)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 26 2013
a(n) = (-2)^n*hypergeom([2, -n], [], 1/2) = 4*(-2)^n*(1 - 2*hypergeom([1, -n-3], [], 1/2))/(n^2+3*n+2) = (4*(-2)^n + Gamma(n+4, -2)*exp(-2))/(n^2+3*n+2). - Vladimir Reshetnikov, Oct 07 2016
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EXAMPLE
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G.f. = 1 + 2*x^2 + 4*x^3 + 24*x^4 + 128*x^5 + 880*x^6 + 6816*x^7 + ...
Since a(1) = 0, then, for n = 2, we have a(2) = -(-2)^3/4 = 2; further, for n = 3, we find a(3) = (3*6/5)*2 - (-2)^4/5 = 36/5 - 16/5 = 4. - Vladimir Shevelev, Apr 01 2010
a(4) = 24 because there are 6 derangements with one 4-cycle with 2^1 ways to color each derangement and 3 derangements with two 2-cycles with 2^2 ways to color each derangement. - Michael Somos, Jan 19 2011
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MAPLE
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seq(simplify(KummerU(-n, -n-1, -2)), n = 0..24); # Peter Luschny, May 10 2022
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MATHEMATICA
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Range[0, 20]! CoefficientList[Series[Exp[-2 x]/(1 - x)^2, {x, 0, 20}], x]
Table[(-2)^n HypergeometricPFQ[{2, -n}, {}, 1/2], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 07 2016 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, n! * polcoeff( exp( -2 * x + x * O(x^n) ) / ( 1 - x )^2, n ) )} /* Michael Somos, Jan 19 2011 */
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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Definition via e.g.f. from Eric Rains, Mar 15 2004
Changed the offset and terms to correspond to e.g.f, Michael Somos, Jan 19 2011
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STATUS
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approved
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